Abstract
The terminal backup problems [Anshelevich and Karagiozova, 2011] form a class of network design problems: Given an undirected graph with a requirement on terminals, the goal is to find a minimum cost subgraph satisfying the connectivity requirement. The nodeconnectivity terminal backup problem requires a terminal to connect other terminals with a number of nodedisjoint paths. This problem is not known whether is NPhard or tractable. Fukunaga (2016) gave a 4/3approximation algorithm based on LProunding scheme using a general LPsolver.
In this paper, we develop a combinatorial algorithm for the relaxed LP to find a halfintegral optimal solution in O(mlog (mUA)⋅ MF(kn,m+k²n)) time, where m is the number of edges, k is the number of terminals, A is the maximum edgecost, U is the maximum edgecapacity, and MF(n',m') is the time complexity of a maxflow algorithm in a network with n' nodes and m' edges. The algorithm implies that the 4/3approximation algorithm for the nodeconnectivity terminal backup problem is also efficiently implemented. For the design of algorithm, we explore a connection between the nodeconnectivity terminal backup problem and a new type of a multiflow, called a separatelycapacitated multiflow. We show a minmax theorem which extends Lovász  Cherkassky theorem to the nodecapacity setting. Our results build on discrete convex analysis for the nodeconnectivity terminal backup problem.
BibTeX  Entry
@InProceedings{hirai_et_al:LIPIcs:2020:12472,
author = {Hiroshi Hirai and Motoki Ikeda},
title = {{NodeConnectivity Terminal Backup, SeparatelyCapacitated Multiflow, and Discrete Convexity}},
booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
pages = {65:165:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771382},
ISSN = {18688969},
year = {2020},
volume = {168},
editor = {Artur Czumaj and Anuj Dawar and Emanuela Merelli},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12472},
URN = {urn:nbn:de:0030drops124725},
doi = {10.4230/LIPIcs.ICALP.2020.65},
annote = {Keywords: terminal backup problem, nodeconnectivity, separatelycapacitated multiflow, discrete convex analysis}
}
Keywords: 

terminal backup problem, nodeconnectivity, separatelycapacitated multiflow, discrete convex analysis 
Collection: 

47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) 
Issue Date: 

2020 
Date of publication: 

29.06.2020 